Subgroups — C₆₇×#10847 #30690098

Order 17152.

898 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#423×C₆₇	C₂	1	2	yes	maximal
2	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#343×C₆₇	C₂	1	4	yes	maximal
4	C₂²×M₁₆×C₆₇	C₂²	1	4	yes	—
5	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
6	C₂²×C₄²×C₆₇	C₄	1	2	yes	—
7	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
8	#423×C₆₇	—	2	4	no	—
9	#423×C₆₇	C₄	1	8	yes	complemented
10	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
11	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
12	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
13	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
14	C₂×#95×C₆₇	C₄	1	2	yes	—
15	#343×C₆₇	—	2	8	no	—
16	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
18	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₆₇	—	2	4	no	—
21	C₂×M₁₆×C₆₇	—	2	8	no	—
22	C₂×M₁₆×C₆₇	—	2	8	no	—
23	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₆₇	—	2	4	no	—
25	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
26	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
27	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
28	C₂×C₄⋊C₄×C₆₇	Q₈	1	4	yes	—
29	C₂×C₄²×C₆₇	—	2	2	no	—
30	C₂×C₄²×C₆₇	—	2	2	no	—
31	C₂×C₄²×C₆₇	—	4	4	no	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₆₇	D₄	1	2	yes	—
34	C₂×C₄²×C₆₇	D₄	1	2	yes	—
35	C₂×C₄²×C₆₇	D₄	1	2	yes	—
36	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
37	C₄⋊C₈×C₆₇	—	2	8	no	—
38	C₄⋊C₈×C₆₇	—	2	8	no	—
39	#95×C₆₇	—	2	4	no	—
40	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	—	2	2	no	—
42	C₂²×C₄×C₆₇	—	2	2	no	—
43	C₂²×C₄×C₆₇	—	2	4	no	—
44	C₂²×C₄×C₆₇	—	2	4	no	—
45	C₂²×C₄×C₆₇	—	4	2	no	—
46	C₂²×C₄×C₆₇	—	4	4	no	—
47	C₂²×C₄×C₆₇	—	4	4	no	—
48	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
49	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
55	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₆₇	#41	1	4	yes	—
63	M₁₆×C₆₇	—	2	16	no	—
64	M₁₆×C₆₇	—	4	4	no	—
65	M₁₆×C₆₇	—	4	4	no	—
66	C₂×C₈×C₆₇	—	2	8	no	—
67	C₂×C₈×C₆₇	—	4	8	no	—
68	C₄⋊C₄×C₆₇	—	2	8	no	—
69	C₄⋊C₄×C₆₇	—	2	8	no	—
70	C₄²×C₆₇	—	4	4	no	—
71	C₄²×C₆₇	—	4	4	no	—
72	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
73	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
74	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
75	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
76	C₂³×C₆₇	—	2	1	no	—
77	C₂³×C₆₇	—	2	1	no	—
78	C₂³×C₆₇	—	2	2	no	—
79	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
80	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
83	C₂³×C₆₇	#117	1	2	yes	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	2	4	no	—
88	C₂×C₄×C₆₇	—	2	4	no	—
89	C₂×C₄×C₆₇	—	2	8	no	—
90	C₂×C₄×C₆₇	—	4	2	no	—
91	C₂×C₄×C₆₇	—	4	2	no	—
92	C₂×C₄×C₆₇	—	4	8	no	—
93	C₂×C₄×C₆₇	—	4	4	no	—
94	C₂×C₄×C₆₇	—	4	4	no	—
95	C₂×C₄×C₆₇	—	4	8	no	—
96	C₂×C₄×C₆₇	#95	1	2	yes	—
97	C₂×C₄×C₆₇	#95	1	4	yes	—
98	C₂×C₄×C₆₇	#95	1	2	yes	—
99	C₂×C₄×C₆₇	#100	1	2	yes	—
100	C₂×C₄×C₆₇	#101	1	2	yes	—
101	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
104	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
105	C₂×C₄×C₆₇	#121	1	4	yes	—
106	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
107	C₂×C₄×C₆₇	#124	1	2	yes	—
108	C₂×C₄×C₆₇	#125	1	2	yes	—
109	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
111	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
112	C₈×C₆₇	—	4	8	no	—
113	C₂²×C₆₇	—	2	2	no	—
114	C₂²×C₆₇	—	2	2	no	—
115	C₂²×C₆₇	—	2	2	no	—
116	C₂²×C₆₇	—	2	2	no	—
117	C₂²×C₆₇	—	2	1	no	—
118	C₂²×C₆₇	—	2	1	no	—
119	C₂²×C₆₇	—	2	1	no	—
120	C₂²×C₆₇	—	2	1	no	—
121	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
122	C₂²×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
123	C₂²×C₆₇	#380	1	2	yes	—
124	C₂²×C₆₇	#382	1	1	yes	characteristic
125	C₂²×C₆₇	#384	1	1	yes	characteristic
126	C₂²×C₆₇	C₂×#100	1	1	yes	characteristic
127	C₂²×C₆₇	C₂×#101	1	1	yes	characteristic
128	C₂²×C₆₇	#413	1	2	yes	—
129	C₄×C₆₇	—	2	2	no	—
130	C₄×C₆₇	—	2	4	no	—
131	C₄×C₆₇	—	4	8	no	—
132	C₄×C₆₇	#343	1	4	yes	complemented
133	C₄×C₆₇	#389	1	4	yes	—
134	C₄×C₆₇	#390	1	2	yes	—
135	C₄×C₆₇	#391	1	2	yes	—
136	#10847	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂×C₆₇	—	2	2	no	—
138	C₂×C₆₇	—	2	1	no	—
139	C₂×C₆₇	—	2	1	no	—
140	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
141	C₂×C₆₇	C₂×#343	1	1	yes	characteristic
142	C₂×C₆₇	#1742	1	1	yes	characteristic
143	C₂×C₆₇	#1800	1	2	yes	—
144	C₂×C₆₇	#1801	1	2	yes	—
145	C₂×#423	C₂×C₆₇	1	2	yes	—
146	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
147	C₂×#343	C₂×C₆₇	1	4	yes	—
148	C₆₇	#10847	1	1	yes	characteristic, direct product factor, Sylow
149	C₂²×M₁₆	C₂²×C₆₇	1	4	yes	—
150	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
151	C₂²×C₄²	C₄×C₆₇	1	2	yes	—
152	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
153	#423	—	2	4	no	—
154	#423	C₄×C₆₇	1	8	yes	complemented
155	C₂×C₄⋊C₈	—	2	2	no	—
156	C₂×C₄⋊C₈	—	2	2	no	—
157	C₄×C₄⋊C₄	—	2	2	no	—
158	C₄×C₄⋊C₄	—	2	2	no	—
159	C₂×#95	C₄×C₆₇	1	2	yes	—
160	#343	—	2	8	no	—
161	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
162	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
163	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
164	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	—	2	8	no	—
167	C₂×M₁₆	—	2	8	no	—
168	C₂×M₁₆	C₂×C₄×C₆₇	1	8	yes	—
169	C₂²×C₈	—	2	4	no	—
170	C₂×C₄⋊C₄	—	2	4	no	—
171	C₂×C₄⋊C₄	—	2	4	no	—
172	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
173	C₂×C₄⋊C₄	Q₈×C₆₇	1	4	yes	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	—	4	4	no	—
177	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
178	C₂×C₄²	D₄×C₆₇	1	2	yes	—
179	C₂×C₄²	D₄×C₆₇	1	2	yes	—
180	C₂×C₄²	D₄×C₆₇	1	2	yes	—
181	C₂×C₄²	Q₈×C₆₇	1	2	yes	—
182	C₄⋊C₈	—	2	8	no	—
183	C₄⋊C₈	—	2	8	no	—
184	#95	—	2	4	no	—
185	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	C₄²×C₆₇	1	2	yes	—
194	C₂²×C₄	C₄²×C₆₇	1	2	yes	—
195	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
196	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
197	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
198	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
199	C₂²×C₄	C₄⋊C₄×C₆₇	1	8	yes	—
200	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
201	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
202	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
205	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
207	C₂²×C₄	#41×C₆₇	1	4	yes	—
208	M₁₆	—	2	16	no	—
209	M₁₆	—	4	4	no	—
210	M₁₆	—	4	4	no	—
211	C₂×C₈	—	2	8	no	—
212	C₂×C₈	—	4	8	no	—
213	C₄⋊C₄	—	2	8	no	—
214	C₄⋊C₄	—	2	8	no	—
215	C₄²	—	4	4	no	—
216	C₄²	—	4	4	no	—
217	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
218	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
219	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
220	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
221	C₂³	—	2	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	2	no	—
224	C₂³	C₂×C₄²×C₆₇	1	1	yes	characteristic
225	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
226	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
227	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
228	C₂³	#117×C₆₇	1	2	yes	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	#95×C₆₇	1	2	yes	—
242	C₂×C₄	#95×C₆₇	1	4	yes	—
243	C₂×C₄	#95×C₆₇	1	2	yes	—
244	C₂×C₄	#100×C₆₇	1	2	yes	—
245	C₂×C₄	#101×C₆₇	1	2	yes	—
246	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
247	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
248	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
249	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
250	C₂×C₄	#121×C₆₇	1	4	yes	—
251	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
252	C₂×C₄	#124×C₆₇	1	2	yes	—
253	C₂×C₄	#125×C₆₇	1	2	yes	—
254	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
255	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
256	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
257	C₈	—	4	8	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	C₂×#95×C₆₇	1	1	yes	characteristic
267	C₂²	C₄×C₄⋊C₄×C₆₇	1	2	yes	—
268	C₂²	#380×C₆₇	1	2	yes	—
269	C₂²	#382×C₆₇	1	1	yes	characteristic
270	C₂²	#384×C₆₇	1	1	yes	characteristic
271	C₂²	C₂×#100×C₆₇	1	1	yes	characteristic
272	C₂²	C₂×#101×C₆₇	1	1	yes	characteristic
273	C₂²	#413×C₆₇	1	2	yes	—
274	C₄	—	2	2	no	—
275	C₄	—	2	4	no	—
276	C₄	—	4	8	no	—
277	C₄	#343×C₆₇	1	4	yes	complemented
278	C₄	#389×C₆₇	1	4	yes	—
279	C₄	#390×C₆₇	1	2	yes	—
280	C₄	#391×C₆₇	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	2	1	no	—
283	C₂	—	2	1	no	—
284	C₂	C₆₇×C₄⋊#95	1	1	yes	characteristic
285	C₂	C₂×#343×C₆₇	1	1	yes	characteristic
286	C₂	C₆₇×#1742	1	1	yes	characteristic
287	C₂	C₆₇×#1800	1	2	yes	—
288	C₂	C₆₇×#1801	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
9. #423×C₆₇	C₄	semidirect
132. C₄×C₆₇	#343	semidirect
136. #10847	C₆₇	direct (N × H)
148. C₆₇	#10847	direct (N × H)
154. #423	C₄×C₆₇	semidirect
277. C₄	#343×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#423×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×#343×C₆₇	2
#10847	67

Subgroups — C67×#10847 #30690098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10847 #30690098